Neural Network that Predicts Game State Based on Actions
You'll probably want to start out with "Action-Conditional Video Prediction using Deep Networks in Atari Games" (arXiv link: https://arxiv.org/abs/1507.08750). That's from 2015 though, I'm sure there have been lots of other interesting developments since then. This paper may still be a good starting point though, and provide you with the correct terminology to plug into google / google scholar to find more recent papers that build on top of this. Google scholar also provides functionality to automatically find papers that cite this one (interesting recent papers will probably cite this one).

As an additional point, you may want to reconsider your desired output. Given a current state-image and action, it may be easier to train a network to predict only the change in image (i.e., predicting NEW_IMAGE - OLD_IMAGE), rather than predicting the full image. You can then always still manually reconstruct the predicted new image simply by adding that output to the old image again. I'm quite sure I've seen this being done in a more recent paper too, but don't remember exactly the title / authors.